\int \frac{dx}{x+\sqrt{x}}=\begin{Bmatrix}
t=\sqrt{x}\\dx=2tdt 

\end{Bmatrix}=\int \frac{2t}{t^{2}+t}dt=2\int \frac{dt}{t+1}=2ln\left | t+1 \right |+C=2ln\left | \sqrt{x}+1 \right |+C